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An inventory system with service facility and finite orbit size for feedback customers

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Abstract

In this article, we consider a continuous review inventory system with service facility consisting of finite waiting hall and a single server. The customers arrive according to a Poisson process and any arriving customer, who finds the waiting hall is full, is considered to be lost. The individual customer’s unit demand is satisfied after a random time of service, which is assumed to be exponential distribution. After a customer is served, he/she will decide either to join the retrial group, which is of finite size, called orbit, for another service or leave the system according to a Bernoulli trial. The customers in the orbit, called feedback customers, compete for service according to classical retrial policy. The service times for these feedback customers are assumed to be independent and exponential distribution. The inventory is replenished according to an (s, S) inventory policy, and the replenishing times are assumed to be exponential. The joint probability distribution of the inventory level, the number of customers in the orbit, number of customers in the waiting hall and the status of the server is obtained in the steady state. Some important system performance measures in the steady state are derived, and the long-run total expected cost rate is also calculated. We have derived the Laplace-Stieljes transforms of waiting time distribution of both primary as well as the feedback customers. The sensitivity analysis has been carried out to examine the effect of different parameters and cost in the system.

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References

  1. Abate J., Whitt W.: Numerical inversion of laplace transforms of probability distributions ORSA. J. Comput. 7, 36–43 (1995)

    Google Scholar 

  2. Ali, O. M. E., Netus, M. E.: A service system with two stages of waiting and feedback customers. J. Appl. Probab. 21, 404–413 (1984)

    Article  Google Scholar 

  3. Arivarignan, G., Sivakumar, B.: Inventory system with renewal demands at service facilities. In: Srinivasan, S.K., Vijayakumar, A. (eds.) Stochastic Point Processes, pp 108–123. Narosa Publishing House, New Delhi (2003)

    Google Scholar 

  4. Arivarignan, G., Yadavalli, V.S.S., Sivakumar, B.: A perishable inventory system with multi-server service facility and retrial customers. In: Ravichandran, N. (ed.) Management Science and Practice, pp 3–27. Allied Publishers pvt. Ltd., New Delhi (2008)

    Google Scholar 

  5. Austin, J. Lemoine: Some Limiting Results for a Queueing Model with Feedback. Sankhya: The Indian Journal of Statistic, series A 36, 293–304 (1974)

    Google Scholar 

  6. Berman, O., Kaplan, E. H., Shimshak, D. G.: Deterministic approximations for inventory management at service facilities. IIE Trans. 25, 98–104 (1993)

    Article  Google Scholar 

  7. Berman, O., Kim, E.: Stochastic inventory policies for inventory management of service facilities. Stoch. Model. 15, 695–718 (1999)

    Article  Google Scholar 

  8. Berman, O, Sapna, K. P.: Inventory management at service facilities for systems with arbitrarily distributed service times. Stoch. Model. 16, 343–360 (2000)

    Article  Google Scholar 

  9. Choi, B. D., Kim, B., Choi, S. H.: On the M/G/1 Bernoulli feedback queue with multi class customers. Comput. Oper. Res. 27, 269–286 (2000)

    Article  Google Scholar 

  10. Gilles, R.D., Disney, R.L.: Queues with instantaneous feedback. Manag. Sci. 24, 168–180 (1977)

    Article  Google Scholar 

  11. He, Q-M, Jewkes, E. M., Buzacott, J.: An Efficient algorithm for computing the optimal replenishment policy for an inventory-production system. In: Alfa, A., Chakravarthy, S. (eds.) Advances in Matrix Analytic Methods for Stochastic Models, pp 381–402. Notable Publications, New Jersey (1998)

    Google Scholar 

  12. Krishnamoorthy, A., Lakshmy, B., Manikandan., R.: A Survey of inventory models with positive service time. OPSEARCH 48, 153–169 (2011)

    Article  Google Scholar 

  13. Krishna Kumar, B., Pavai Madheswari, S., Vijayakumar, A.: The M/G/1 retrial queue with feedback and starting failures. Appl. Math. Model. 26, 1057–1075 (2002)

    Article  Google Scholar 

  14. Krishna Kumar, B., Raja, J.: On multiserver feedback retrial queues with balking and control retrial rate. Ann. Oper. Res. 141, 211–232 (2006)

    Article  Google Scholar 

  15. Krishna Kumar, B., Rukmani, R., Thangaraj, V.: On multiserver feedback retrial queue with finite buffer. Appl. Math. Model. 33, 2062–2083 (2009)

    Article  Google Scholar 

  16. Lee, Y.W.: The M/G/1 feedback retrial queue with two types of customers. Bull. Korean Math. Soc. 42, 875–887 (2005)

    Article  Google Scholar 

  17. Manuel, P., Sivakumar, B., Arivarignan, G.: A perishable inventory system with service facilities, map arrivals and ph-service times. J. Syst. Sci. Syst. Eng. 16, 62–73 (2007)

    Article  Google Scholar 

  18. Manuel, P., Sivakumar, B., Arivarignan, G.: A perishable inventory system with service facilities and retrial customers. Comput. Ind. Eng. 54, 484–501 (2008)

    Article  Google Scholar 

  19. Schwarz, M., Sauer, C., Daduna, H., Kulik, R., Szekli, R.: M/M/1 queueing systems with inventory. Queueing Syst. 54, 55–78 (2006)

    Article  Google Scholar 

  20. Sivakumar, B., Arivarignan, G.: A perishable inventory system at service facilities with negative customers. Int. J. Inf. Manag. Sci. 17, 1–18 (2006)

    Google Scholar 

  21. Takacs, L.: A single server queue with feedback. Bell Syst. Tech. J. 42, 505–519 (1963)

    Article  Google Scholar 

  22. Thangaraj, V., Santhakumaran, A.: Sojourn times in queues with a pair of instantaneous independent Bernoulli feedback. Optimization 29, 281–294 (1994)

    Article  Google Scholar 

  23. Thangaraj, V., Vanitha, S.: A single server M/G/1 feedback queue with two type of service having general distribution. Int. Math. Forum 5, 15–13 (2010)

    Google Scholar 

  24. Yadavalli, V.S.S., Sivakumar, B., Arivarignan, G., Adetunji, O.: A Multi-server perishable inventory system with negative customer. Comput. Ind. Eng. 61, 254–273 (2011)

    Article  Google Scholar 

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Acknowledgments

The authors wish to thank the anonymous referees for their comments, which significantly improved the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematics, Madurai Kamaraj University, Madurai, India

    M. Amirthakodi & B. Sivakumar

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  1. M. Amirthakodi
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  2. B. Sivakumar
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Correspondence to M. Amirthakodi.

Additional information

The research is supported by University Grants Commission - Rajiv Gandhi National Fellowship, NEW DELHI, research award No. F. 14-2(SC)/2010 (SA-III).

The research is supported by the National Board for Higher Mathematics, INDIA, research award 2/48(4)/2011/-R&D III/4721.

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Amirthakodi, M., Sivakumar, B. An inventory system with service facility and finite orbit size for feedback customers. OPSEARCH 52, 225–255 (2015). https://doi.org/10.1007/s12597-014-0182-5

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